By Thomas Ernst

To date, the theoretical improvement of q-calculus has rested on a non-uniform foundation. in most cases, the cumbersome Gasper-Rahman notation was once used, however the released works on q-calculus seemed various looking on the place and by means of whom they have been written. This confusion of tongues not just complex the theoretical improvement but additionally contributed to q-calculus ultimate a overlooked mathematical box. This booklet overcomes those difficulties via introducing a brand new and engaging notation for q-calculus in accordance with logarithms.For example, q-hypergeometric services are actually visually transparent and simple to track again to their hypergeometric mom and dad. With this new notation it's also effortless to work out the relationship among q-hypergeometric features and the q-gamma functionality, whatever that before has been overlooked.

The booklet covers many issues on q-calculus, together with targeted capabilities, combinatorics, and q-difference equations. except a radical evaluation of the historic improvement of q-calculus, this ebook additionally offers the domain names of recent physics for which q-calculus is acceptable, reminiscent of particle physics and supersymmetry, to call only a few.

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**Extra resources for A Comprehensive Treatment of q-Calculus**

**Sample text**

However, not all practitioners of the AmericanAustrian School wrote books on their subject—the largest part of their contribution is contained in their lectures and articles. Vandiver and Gould collected card-files on articles about Bernoulli numbers; this work was continued by Karl Dilcher and made available on the Web. It is also possible that Carlitz was a student of Schwatt at University of Pennsylvania, Philadelphia. Carlitz was born in Philadelphia, and the two have worked on similar mathematical topics.

4. 5. 6. 7. Jacobi theta functions and elliptic functions of one variable. Riemann Theta functions of one or more variables or Abelian functions. Weierstraß elliptic functions and sigma functions. The Glaisher-Neville-School for Jacobi elliptic functions. The q -School. The Heine function School. The Weierstraß-Mellin School of Gamma functions and hypergeometric functions. 8. The Italian elliptic function School. School 1 is about the Jacobi theta functions, so called after the German mathematician Carl Gustav Jacob Jacobi.

Vandiver and Gould collected card-files on articles about Bernoulli numbers; this work was continued by Karl Dilcher and made available on the Web. It is also possible that Carlitz was a student of Schwatt at University of Pennsylvania, Philadelphia. Carlitz was born in Philadelphia, and the two have worked on similar mathematical topics. Schwatt, who became PhD in 1893 in Philadelphia, remained in this city during his whole career (1897–1928). We will come back to the q-analogues of the Schwatt formulas in Chapter 5, where q-Stirling numbers are discussed.